The victim's father, Neville, said he was "full of joy and relief that two of my son's killers had been convicted" as a statement was read out by his solicitors.

Mr Lawrence said he was conscious there were five or six attackers that night and that he didn't think he would "be able to rest until they are brought to justice".

Stephen Lawrence's friend Duwayne Brooks, who was with him when he was killed, welcomed the verdicts, writing on Twitter: "Some JUSTICE at last."

David Cameron said: "In the 19 years since his murder, Stephen Lawrence's family has fought tirelessly for justice. Today's verdict cannot ease the pain of losing a son. But, for Doreen and Neville Lawrence, I hope that it brings at least some comfort after their years of struggle."

The police chief leading the Lawrence murder inquiry praised the victim's parents for bringing major changes "in society as a whole".

Cressida Dick, acting deputy commissioner, said Scotland Yard was "very pleased for the family and those who loved Stephen" as she welcomed Tuesday's verdict.

"There have been several investigations over the last 18 years during which the Lawrence family have campaigned tirelessly for justice," she said.

Former mayor of London Ken Livingstone said Mr and Mrs Lawrence had "altered the terms of how we think about racism".

"It is extremely moving that the Lawrence family has finally secured a measure of justice," he said.

"This has been a momentous and long-overdue verdict. In the course of their dignified campaign for justice, Doreen and Neville Lawrence have changed many lives, and altered the terms of how we think about racism in our country.

"Institutions and individuals, including, of course, the Metropolitan police, had to re-examine how they work.

"At the heart of this is the case of a young man killed because he was black. Our thoughts today should be first and foremost with Stephen's family."

Michael Mansfield QC, who represented Lawrence's parents at his inquest, hailed the "huge dignity and persistence" they had shown in seeking justice.

He added: "It is a component in changing the public perception of these cases. This is just another milestone for the family in a very long journey.

"It is a real tribute to the family that they have fought so long to bring about change."

Paul Anderson-Walsh, chief of the Stephen Lawrence Charitable Trust, hailed Doreen Lawrence for being "a metaphor for the very best of British values".

He said: "Throughout the 18-year ordeal, the Lawrence family's desire has been the pursuit of justice; today, justice was served. As one door closes, they are determined to make sure that other doors open."

Anderson-Walsh added: "Stephen Lawrence's murder leaves in its wake a changed criminal justice landscape, but it is a change in the social justice topography that the Lawrence family hopes will be Stephen Lawrence's lasting legacy.

"The greatest desire of the Lawrence family is to give to others what was so cruelly snatched from Stephen – a chance to fulfil their potential in life.

"Through the trust that bears his name, Doreen Lawrence and those who work for and with the trust are helping to grow a fairer society by watering the seeds of ambition, one 'Stephen' at a time.

"Whilst Stephen's death has become a metaphor for individual and institutional racism, Stephen's mother, the inspirational Doreen Lawrence OBE, is a metaphor for the very best of British values: dignity, courage and sheer tenacity."

Clive Efford, Labour MP for Eltham, where the attack took place, said he hoped the police investigation would continue if officers believe there were others involved.

Efford told the BBC: "We have all seen the publicity and the suggestion of who those people might be.

"It has always been suggested it was five, possibly six, suspects and if they have any other evidence that will allow further prosecutions in the future I hope we will see those because justice must be done in this case.

"We are part of the way there today and there are others that were involved that must face justice as well."

He added: "The new technology and science that is available to us means that convictions like this are possible.

"That was the reason double jeopardy was introduced, because the technology was allowing us to go back and review some previous court decisions and in the case of murder it was decided that we should be allowed to use that new science."

Forensic scientists who unearthed new evidence leading to the conviction said they were "hugely" proud.

LGC forensic services, a private firm, said its work in securing fibres from clothing and blood fragments in the packaging had been "painstaking and meticulous".

Steve Allen, managing director of LGC Forensics, said: "I'm extremely proud of the work that LGC's forensic scientists did on this case. Persistence, meticulous science and innovation can help convict criminals years after they committed the crime.

"This case shows that the key to successful forensics is to assume nothing - which is all the more important in historic cases like the murder of Stephen Lawrence."

Bevan Powell, chairman of the Metropolitan Black Police Association, said the verdicts "bring a sense of justice to a long overdue prosecution".

He added: "The Lawrences' persistence brought about a social consciousness which questioned and challenged the then existing inadequacies and failures inherent within the police and wider justice system to deliver fairness in cases which involved race and minority communities."

Charles Crichlow, president of the association, added: "These guilty verdicts represent a triumph for the Lawrence family's determination and despite the failings of the initial police investigation they represent also a triumph of justice."

The Labour leader, Ed Miliband, said he hoped the verdict would help bring closure for the Lawrence family.

He said: "The murder of Stephen Lawrence was not only a tragedy for a talented young man and his family, it was a wake-up call to all of us who believe Britain is – and always must be – a country where everyone is shown respect irrespective of race, culture or faith.

"I am proud that 18 years on from Stephen's murder, Britain is a much more tolerant and open country. We have new laws to prevent the stirring of racial hatred, while following the Macpherson report, reforms were made to ensure that our public institutions are better representative and responsive to all the people.

"But we must never allow ourselves to become complacent about the threat of racism. And we must continue to confront it, in all its forms."

They are responsible for the conviction of the killers in some of Britain's highest-profile murders, including those of Stephen Lawrence, Damilola Taylor and Rachel Nickell. What's the secret of their success?